## Did I say that we were Building a Rocket?

by Robert Brand. **No we haven’t**, but here is the buzz – we are developing significant rocket technology.

It was ThunderStruck team member David Galea that headed his email with “Greetings Fellow Rocketeers” and it may stick because ThunderStruck is building rocket technology. We may be building more rockets later but right now we are specifically building a booster for a bigger rocket. A booster that could make it to space all by itself with a ThunderStruck suborbital winged craft as the payload (mounted right on top of the thruster). The rocket will be configured as a sounding rocket – not orbital. The picture (above right) is a similar craft, but a way bigger craft, on top of a bigger rocket. Non the less it will look similar.

This will take years to build and it may result in a static test fire in the Australian desert in the next year or two depending on financing. None the less, it will be an amazing opportunity for a small company to gain considerable traction in the rocket building field.

The info here is a basic format that hopefully high school students can understand

### Rockets and Maths

Mathematics is essential in building space equipment, space craft and navigating in space to mention a tiny bit. Without maths, rockets would explode from over-pressure or fail to get to space because we over-engineered it and it was too heavy to be a work horse.

The image at right is a basic configuration. Solid fuel with an air core and a thrust and nozzle at the bottom. Looks simple, but the maths have to be done first to get an estimation of the pressure we can expect and the strength of the tank and the weight of the tank with different metals. note that as the fuel burns down from the inside towards the metal of the tank, the area burning is greater and the pressure thus increases in a big way. You can change the fuel configuration to burn slower or have less thrust, but that could change simplicity of equation below so we will assume that the fuel is the same for the entire burn. That has been done and we came up with two limits on the mass that we can now work with. The optimum design will be in the middle somewhere.

After putting a rough design on the table with a mass of 2,000Kg fully fueled, we managed to get to space with a big payload and a coasting altitude of 150Km or more. This was with a speed of 1.5Km (or more) per second at the 30 second burn when the fuel is exhausted.

A second design with 3,000Kg mass fully fueled only managed a bit less than 25km altitude. The optimum booster, configured as a sounding rocket lies somewhere in between. The next part of the work is to consider the options. That is:

- Do we use more thrust and increase the tank and nozzle pressure?
- We we increase the fuel load and mass?
- Do we reduce the fuel load and mass?
- Do we change the fuel and increase the pressure and even the burn time?
- Do we reduce the mass of the payload (250Kg in this initial desktop design?
- Do we reduce the mass of the rocket?

These are just a few of the options, but how do we calculate these things – Mathematics of course.

Below are the maths for the heavier second design that only got to under 25Km configures as a rocket. It would have made a poor booster.

NOTE: this is a simple bit of maths for model rockets, but it applies to the bigger ones too. It is not the whole deal, but will give a good estimate for the first pass.

David Galea’s maths for the second configuration performance:

**ThunderStruck Rocket Flight Profile – Estimated Calculations**

There are three basic equations to find the peak altitude for the rocket

916*Max velocity v, the velocity at burnout = q*[1-exp(-x*t)] / [1+exp(-x*t)] =*

13,191.684 m*Altitude reached at the end of boost = [-M / (2*k)]*ln([T – M*g – k*v^2] / [T – M*g]) =*11,515.9877 m*Additional height achieved during coast = [+M / (2*k)]*ln([M*g + k*v^2] / [M*g]) =*

**Total Height Achieved = 24,707.67 Km**

All the terms in these equations are explained below on the method for using the equations.

- Compute Some Useful Terms
- Find the
**mass M**of your rocket in kilograms (kg): 2950kg - Find the
**area A**of your rocket cross-section in square meters (m^2): 0.342m^2 - Note that the
**wind resistance force**= 0.5 * rho*Cd*A * v^2, where

is density of air = 1.2 kg/m^3*rho*

is the drag coefficient of your rocket which is around 0.75 for a model rocket shape.*Cd*

is the velocity of the rocket. You*v***don’t**calculate this drag force, though, since you don’t know what “v” is yet. What you**do**need is to lump the wind resistance factors into one coefficient k:

k = 0.5*rho*Cd*A = 0.5*1.2*0.75*A = 0.1539 - Find the
**impulse I and thrust T**of the engine for your rocket. I= 3907501 Ns , T= 118841.27 Ns - Compute the
**burn time t**for the engine by dividing impulse I by thrust T:

t = I / T = 3907501 / 118841.27 = 32.88 seconds - Note also – the
**gravitational force**is equal to M*g, or the mass of the rocket times the acceleration of gravity (g). The value of g is a constant, equal to 9.8 meters/sec/sec. This force is the same as the weight of the rocket in newtons.

- Find the
- Compute a couple of terms, I call them “q” and “x”
- q = sqrt([T – M*g] / k) = sqrt([118841.27 – 2950 * 9.8] / 0.1539) = 764.427
- x = 2*k*q / M = 2 * 0.1539 * 764.427 / 2950 = 0.079759536

- Calculate velocity at burnout (max velocity, v), boost phase distance yb, and coast phase distance yc (you will sum these last two for total altitude).
- v = 764.427*[1-exp(-0.079759536*32.88)] / [1+exp(-0.079759536*32.88)] = 660.916
- yb = [-2950 / (2*0.1539)]*ln([118841.27 – 2950 *9.8 – 0.1539*660.916^2] / [118841.27 – 2950 *9.8]) = 13191.684
- yc = [+2950 / (2*0.1539)]*ln([2950 *9.8 + 0.1539*660.916^2] / [2950 *9.8]) = 11515.9877

David says: *I have double checked my calculations with wolfram alpha (https://www.wolframalpha.com) with the same results.*

Well fellow Rocketeers, we will continue to let you know about our big adventure with things that could “go BANG” as we develop our technology.

The Screen shot at right is a basic program that you can get for free or you can buy a more professional version for model rocket hobbyists. None the less it is fine for early desktop modeling.

We will keep you in touch with the professional software that we will eventually choose and use for the serious design phase.

All you students, please get your head down and study maths. We will need to have capable people working in the space sector as Project ThunderStruck becomes an Australian Space staple.